Efficient Flight Control via Mechanical Impedance ... - UCSB ECE

J Intell Robot Syst DOI 10.1007/s10846-013-9928-1

Efficient Flight Control via Mechanical Impedance Manipulation: Energy Analyses for Hummingbird-Inspired MAVs Hosein Mahjoubi · Katie Byl

Received: 31 August 2013 / Accepted: 12 September 2013 © Springer Science+Business Media Dordrecht 2013

Abstract Within the growing family of unmanned aerial vehicles (UAV), flapping-wing micro aerial vehicles (MAV) are a relatively new field of research. Inspired by small size and agile flight of insects and birds, these systems offer a great potential for applications such as reconnaissance, surveillance, search and rescue, mapping, etc. Nevertheless, practicality of these vehicles depends on how we address various challenges ranging from control methodology to morphological construction and power supply design. A reasonable approach to resolving such problems is to acquire further inspiration from solutions in nature. Through modeling synchronous muscles in insects, we have shown that manipulation of mechanical impedance properties at wing joints can be a very efficient method for controlling lift and thrust production in flapping-wing MAVs. In the present work, we describe how this approach can be used to decouple lift/thrust regulation, thus reducing the complexity of flight controller. Although of simple design, this con-

H. Mahjoubi (B) · K. Byl Robotics Laboratory, Department of Electrical and Computer Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA e-mail: [email protected] K. Byl e-mail: [email protected]

troller is still capable of demonstrating a high degree of precision and maneuverability throughout various simulated flight experiments with different types of trajectories. Furthermore, we use these flight simulations to investigate the power requirements of our control approach. The results indicate that these characteristics are considerably lower compared to when conventional control strategies—methods that often rely on manipulating stroke properties such as frequency or magnitude of the flapping motion—are employed. With less power demands, we believe our proposed control strategy is able to significantly improve flight time in future flapping-wing MAVs. Keywords Bio-inspired MAV · Flapping flight · Tunable impedance · Variable stiffness actuator · Maneuverability · Energy efficiency

1 Introduction Originally introduced during World War I [1], Unmanned Aerial Vehicles (UAVs) have experienced a substantial level of growth over the past century; both in military and civilian application domains. Surveillance, reconnaissance, search and rescue, traffic monitoring, fire detection and environment mapping are just a few examples of their many possible applications.

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UAVs can be categorized into three general groups: fixed-wing airplanes, rotary-wing vehicles and flapping wing systems. Each type has different advantages and disadvantages. However, as we go smaller in size to design Micro Aerial Vehicles (MAVs), the limitations of fixed and rotarywing technologies become more apparent; the Reynolds number decreases as the wing surface becomes smaller, resulting in unsteady aerodynamic effects that are still a subject of research. On the other hand, flapping wing designs inspired by birds and insects offer the most potential for high maneuverability at miniaturized scales. Unparalleled dynamics, speed and agility are known traits of insects and hummingbirds that turn them into the most efficient flyers in nature. Furthermore, their ability to adapt with varying weather and environmental conditions is phenomenal; hence mimicking their flight dynamics [2–7] and actuation mechanisms would be a reasonable approach to designing practical MAVs. Note that in contrast to most birds whose wing stroke is largely restricted within the coronal plane, insects—and hummingbirds—employ a wing stroke motion that is mostly contained within the axial plane [5]. This feature enables the creature to control its thrust force and perform tasks such as hovering. A schematic description of wing stroke motion in insect flapping flight has been demonstrated in Fig. 1 [8]. Past research on flapping-wing MAVs has mainly concentrated on development of flapping mechanisms that are capable of producing sufficient lift for levitation. With such platforms available, e.g. Harvard’s Robotic Fly [9], Delf ly [10] and the Entomopter [11], force manipulation and motion control are the next challenges that must be faced. Recent work focused on vertical

Fig. 1 Wing stroke cycle in insect flapping flight (adapted from [8])

acceleration and altitude control has led to some impressive results [12, 13]. Research on forward flight control has been less developed in comparison [14]. In addition to flight control challenges, the small size of flapping-wing MAVs poses considerable limits on available space for actuators, power supplies and any sensory or control modules. Although a major portion of the overall mass is often allocated to batteries [9, 10], the flight time of these vehicles rarely exceeds a few minutes. Thus in terms of efficiency and flight duration, current flapping-wing MAVs are still far inferior to their fixed and rotary-wing counterparts. The majority of works concentrating on force control employ modifications in wing-beat profile to achieve their objective [15, 16]. While this approach proves to be successful in separate control of lift [17] or horizontal thrust [18], simultaneous control of these forces is more challenging. Both forces are directly influenced by velocity of air flow over the wings, a factor that is heavily determined by the rate of wing-beat [19]. Therefore, modifying wing-beat profile will lead to coupling between lift and thrust forces, i.e. an inconvenience for motion control and maneuverability of the vehicle. In contrast to stroke actuation methods, one could investigate possible mechanisms that insects and small birds employ to regulate aerodynamic force during various aerial maneuvers. Control strategies developed based on such mechanisms may further improve controllability and energy efficiency of future MAV prototypes. The Tunable Impedance (TI) method [20] is one such strategy. Inspired by the structure of antagonistic muscle pairs attached to wing joints—also known as synchronous muscles [21]—in insects

Upstroke

1

2

3

4

5

6

7

8

9

10

Downstroke

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and hummingbirds, we have previously shown that mechanical impedance properties of each wing joint can be modeled as a linear torsional spring [22, 23]. Tunable Impedance approach states that through manipulating the properties of these springs, one can modify the passive pitch reversal of each wing. Wing pitch reversal [24] plays a significant role in production and manipulation of lift/thrust through regulating the angle of attack (AoA) of the wing relative to air flow [19]. Thus, TI method can indirectly adjust the mean production of lift and thrust in each individual wing [20, 25]. Although this approach is able to provide a higher degree of controllability compared to traditional wing-beat modification approaches [25, 26], its lift and thrust manipulation procedures are not always completely decoupled [26]. In this work, our first goal is to identify the conditions that may reduce coupling between lift and thrust manipulation when TI approach is employed. We then use these findings to develop a motion controller that is capable of tracking various trajectories in the sagittal plane with a considerably high degree of precision as verified by our simulated experiments. Unlike conventional approaches to flappingwing MAV control, TI does not require any modifications in flapping properties such as frequency or magnitude of the stroke. This suggests that a fixed stroke profile can be used during all operational modes including takeoff, forward acceleration and hovering. Hence, regardless of the maneuver, the MAV will need a relatively constant amount of power to flap its wings. Naturally, this amount should be large enough to provide sufficient lift for levitation. Depending on the desired maneuver, extra power may be required to change the mechanical impedance properties of each wing joint via suitable actuators. Through simulated flight experiments, this work will show that such changes are often very small and do not demand considerable amounts of power. Thus, with a motion controller based on Tunable Impedance approach, we expect that an actual MAV’s power requirements during any maneuver remain almost constant, i.e. close to the amount necessary for hovering. To appreciate the significance of this feature, a new set of flight experiments were simulated. In this group of ex-

periments, wings’ mechanical impedance properties were fixed and aerodynamic forces were controlled through changes in shape and frequency of the flapping profile. Comparing both cases, it is observed that power requirements in the Tunable Impedance method are considerably lower. Therefore, implementation of a similar control approach on an actual MAV may significantly improve efficiency and result in longer flight times. The remainder of this work is organized as follows. Section 2 briefly explains our model for mechanical impedance at the wing joints. Section 3 reviews the employed aerodynamic model for flight simulations. Using this model, we will then describe the underlying concept of Tunable Impedance approach. Lift/thrust decoupling conditions are investigated in Section 4. The dynamic model of MAV and the structure of designed controllers are discussed in Sections 5 and 6, respectively. Results of simulated flight experiments are presented in Section 7. Finally, Section 8 concludes this paper.

2 Mechanical Impedance of the Wing Joint In [23], we reviewed the different types of muscle that most insects employ during flight. Among them, synchronous muscles [21] are the only kind that is directly attached to the wing joint. Thus, they should have a key role in determining mechanical impedance of the wing joint. This mechanical impedance is responsible for regulation of changes in wing’s angle of attack (AoA) in response to aerodynamic torques [20]. Thus, modeling these muscles could further improve our understanding of insect flight. 2.1 Muscle-Joint Connection: Mechanical Model The antagonistic synchronous muscle pairs in Fig. 2 [27] and their connection to the wing joint can be modeled by the mechanical structure illustrated in Fig. 3. The behavior of each muscle can be approximately modeled by a quadratic spring [28], i.e. its output force F is equal to: F (δi ) =

1 2 Aδ + Bδi + C, i = 1, 2 2 i

(1)

J Intell Robot Syst Fig. 2 An illustration of insect’s wing stroke using synchronous muscles: a upstroke and b downstroke. Muscles in the process of contraction are shown in a darker color (adapted from [27])

(a)

(b)

where A, B and C are constant coefficients and δi represent the corresponding change in length of each spring. In [23], using Eq. 1 we have shown that rotational stiffness of the joint in Fig. 3 is derived as: krot = 2R2 (Axc + B)

Hence, the wing joint can be considered as a torsional spring whose stiffness krot and equilibrium point ψ0 are respectively controlled through coactivation and differential activation of an antagonistic synchronous muscle pair. In Section 3 we show how these parameters influence lift and thrust production.

(2)

Here, R is the radius of the joint and xc represents the displacement due to co-activation of antagonistic muscle pair, i.e.: xc =

1 (x1 + x2 ) 2

2.2 Active Impedance Regulation: Power Requirements

(3)

Contraction of synchronous muscle pairs may change both stiffness and equilibrium position of the wing joint. Like all other muscles, these contractions consume energy. In Fig. 3, integration of Eq. 1 with respect to δi shows how much work W is required to change the length of each spring:

The displacement induced by actuator force F Mi on the input end of each spring is shown by xi . The differential displacement of these springs, i.e. xd , is responsible for adjustment of wing’s equilibrium pitch angle ψ0 : xd =

1 (x1 − x2 ) = Rψ0 2

W (δi ) =

(4)

1 3 1 2 Aδ + Bδi + C δi , i = 1, 2 6 i 2

It was also shown [23] how an external torque τext influences the pitch angle ψ of the wing joint:

Since:

τext = krot (ψ − ψ0 )

δ1 = xc − R (ψ − ψ0 )

Fig. 3 A simplified mechanical model of synchronous muscle-joint structure. Each muscle is modeled by a quadratic spring. A force of F Mi pulls the input end of each spring, resulting in the displacement xi . The other sides of both springs are attached to a pulley of radius R which models the joint

(5)

x'

x1

(7)

τext

Δψ

FM1 ψ

z FM2

R x2

(6)

ψ0

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δ2 = xc + R (ψ − ψ0 )

(8)

from Eqs. 2 to 4 the overall work is equal to: 1 W (δ1 ) + W (δ2 ) = 2W (xc ) + krot (ψ − ψ0 )2 (9) 2 The required power for impedance manipulation, i.e. PTI is then calculated by differentiating Eq. 9 with respect to time: PTI

1 = 2F (xc ) x˙ c + k˙ rot (ψ − ψ0 )2 2 − krot (ψ − ψ0 ) ψ˙ 0

(10)

Using Eqs. 1 and 2, it is easy to show that choosing C = 0.5B2 /A: F (xc ) x˙ c =

1 k2 k˙ rot 16A2 R6 rot

(11)

which is then replaced in Eq. 10 to yield: 1 1 k2rot k˙ rot + k˙ rot (ψ − ψ0 )2 − 2 6 8A R 2 krot (ψ − ψ0 ) ψ˙ 0 PTI =

(12)

The last expression allows us to estimate power requirements for manipulation of krot and ψ0 based on instantaneous values of these parameters and their time derivatives. In Section 7, we will use Eq. 12 for power estimation during simulated flight experiments with Tunable Impedance controller.

equations and then proceed to explain the interaction between aerodynamic force and mechanical impedance of the wing joint. Interested readers are encouraged to review [23] for further details on this model. 3.1 Wing Shape and Aerodynamic Forces The components of aerodynamic force at center of pressure (CoP) of the wing are illustrated in Fig. 4a. In this diagram, pitch angle of the wing is represented by ψ. The stroke profile of the wing—i.e. the angle φ in Fig. 4b—is assumed to be sinusoidal: φ = φ0 cos (2π fs t) + β

(13)

where φ0 is the magnitude of stroke and fs represents flapping frequency. β is a slowly varying bias angle and has an insignificant effect on force production [23]. More details on this parameter will be provided in Section 6. Using the quasi-steady-state aerodynamic model of [23] along with Blade Element Method (BEM) [3, 33, 34], the aerodynamic force for the wing shape in Fig. 5 can be estimated by:   F N = 0.0442ρ C N (ψ) φ˙ + 1.3462ψ˙ φ˙ R4W (14)

(a)

(b)

z

y y'

3 The Aerodynamic Model With low Reynolds numbers, aerodynamics of insect flight has become an active field of research over the past several decades [2–7, 29–31]. Various unsteady-state mechanisms have been identified that depending on the species, demonstrate different amounts of contribution to production of aerodynamic forces. Among these phenomena, three mechanisms are the most significant in many flying insects: delayed stall [4], rotational circulation [3, 4], and wake capture [4, 29]. Experimental data have been used to develop mathematical expressions for estimation of forces produced by these mechanisms [4, 19, 32]. We have previously used these results to model the overall aerodynamic force in a hummingbird scale flapping-wing MAV [23]. Here, we will only repeat the final

Downstroke

Wing's Stroke Axis

x'

F

x

F

N

x'

Z

F

D

zCoP

F

F

Y

F

D

Downstroke

X

F

Wing's Pitch Rotation Axis

CoP

T

ψ

-

+

CoP

+

-

φ

Fig. 4 a Wing cross-section (during downstroke) at center of pressure, illustrating the pitch angle ψ. Normal and tangential aerodynamic forces are represented by F N and FT . F Z and F D represent the lift and drag components of the overall force. b Overhead view of the wing/body setup which demonstrates the stroke angle φ. F X and FY are components of F D which represent forward and lateral thrust, respectively

J Intell Robot Syst z

FY = −F D sin φ

Rw

(21)

Wing's Stroke Axis

rCoP y' Wing's Pitch Rotation Axis

CoP

zCoP

Fig. 5 The wing shape used in aerodynamic model and flight simulations. The span of the wing is represented by Rw

FT = 0.0442ρCT (ψ) φ˙ 2 R4W

(15)

where ρ is the density of air and RW represents the span of the wing. As illustrated in Fig. 4a, F N and FT are the normal and tangential components of the overall aerodynamic force with respect to the surface of the wing. C N and CT are angledependent aerodynamic coefficients [4, 23]: C N (ψ) = 3.4 cos ψ  CT (ψ) =

(16) π rad 4 otherwise

0.4 cos2 (2ψ) , |ψ| ≥ 0,

(17)

Lift F Z and drag F D components of the aerodynamic force are calculated as follows: F Z = −F N sin ψ − FT cos ψ

(18)

F D = −F N cos ψ + FT sin ψ

(19)

Forward and lateral thrust, i.e. F X and FY , respectively, are also defined as illustrated in Fig. 4b: F X = F D cos φ

Table 1 Physical characteristics used in Section 3

(20)

Finally, it is important to note that the center of pressure [23] in Fig. 5 is located at: zCoP = 0.0673RW

(22)

rCoP = 0.7221RW

(23)

3.2 The Tunable Impedance Approach: A Review As it was stated in Section 2.1, mechanical impedance properties of the joint tend to oppose any external torque on the wing according to Eq. 5. Aerodynamic force and dynamics of the wing influence the rotation of the wing via the following torque expression: τ = zCoP F N − b ψ ψ˙ − Jψ ψ¨

(24)

in which Jψ and b ψ are the moment of inertia and passive damping coefficient of the wing along its pitch rotation axis, respectively. As illustrated in Fig. 5, zCoP is the distance of CoP from this axis. According to Newton’s Equations of Motion, the left-hand sides of Eqs. 5 and 24 must be equal, hence resulting in: Jψ ψ¨ + b ψ ψ˙ + krot (ψ − ψ0 ) − zCoP F N = 0

(25)

Using the presented aerodynamic model in Section 3.1, Eq. 25 can be numerically solved to calculate changes in ψ and F N for any stroke profile and mechanical impedance values. Other aerodynamic profiles will then be available via Eq. 15 and Eqs. 18–21. The values of wing parameters used in all future simulations are reported in Table 1. Since the employed MAV model is roughly the same size as an average hummingbird, stroke parameters

Symbol

Description

Value

ρ RW Jψ bψ mbody

Air density at sea level Average span of each wing in an average hummingbird Moment of inertia of each wing (pitch rotation) Passive damping coefficient of each wing (pitch rotation) Estimated mass of a two-winged MAV with dimensions of an average hummingbird Standard gravity at sea level

1.28 kg· m−3 8 × 10−2 m 1.564 × 10−8 N· m· s2 5 × 10−6 N· m· s 4 × 10−3 kg

g

9.81 m· s−2

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during hovering flight are set to φ0 = 60◦ and fs = 25 Hz [21]. Similar to a hummingbird, the MAV model has two wings and throughout this work, it is assumed that they have always the same stroke profiles and mechanical impedance properties. Under these assumptions, symmetry dictates

that overall lateral thrust and roll/yaw torques are zero; hence motion is effectively limited within the sagittal plane. This allows us to concentrate only on lift F Z and forward thrust F X of a single wing. Roll and yaw responses of the model are previously investigated in [23, 26]. (d) Pitch Angle of the Wing vs. ψ0

(a) Pitch Angle of the Wing vs. krot 80

80

*

60

krot=k

40

krot=2.5k

20

krot=5k

ψ 0=0

40

ψ 0= -20

20

ψ 0=20

*

(deg)

(deg)

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0 -20

-60

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(f) Instantaneous and Average Thrust: FX

(c) Instantaneous and Average Thrust: FX

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0.02

1

6

2 0 -2 -4

-4 -6

0.01

(e) Instantaneous and Average Lift: FZ

3

0

o

0

(b) Instantaneous and Average Lift: FZ

-1

o

-20

φ

-40

o

60

-6 0

0.01

0.02 Time (s)

0.03

0.04

Fig. 6 a Pitch angle evolution of one wing for ψ0 = 0◦ and different values of krot (k∗ = 1.5 × 10−3 N· m/rad). The corresponding instantaneous (blue) and average (red) lift and forward thrust are plotted in b and c respectively. d Pitch angle evolution of one wing for krot = k∗ and

0

0.01

0.02 Time (s)

0.03

0.04

different values of ψ0 . The corresponding instantaneous (blue) and average (red) lift and forward thrust are plotted in e and f, respectively. All forces are normalized by the estimated weight of the vehicle

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The tunable impedance approach suggests that by modification of impedance properties, one can manipulate pitch profile of either wing while its stroke profile remains unperturbed. In fact, the value of krot primarily determines the range of variations in ψ (Fig. 6a) which in turn affects the magnitude of aerodynamic forces (Fig. 6b, c). Note that when ψ0 = 0◦ , pitch and thrust profiles are odd-symmetric, i.e. the average thrust is close to 0. Introducing a nonzero value of ψ0 perturbs this symmetry (Fig. 6f) by biasing the pitch profile (Fig. 6d). In short, tunable impedance allows us to control overall lift and thrust by manipulating krot and ψ0 , respectively [23]. From Fig. 6, although instantaneous forces are time-dependent, their average values remain constant as long as mechanical impedance properties and stroke parameters are unperturbed. We can use these mean values to further investigate the influence of impedance/stroke parameters in lift/thrust production.

(a)

(b) 0.5 Mean FX (g-Force)

Mean FZ (g-Force)

0.8 0.6 0.4 0.2 0

-0.2 -0.4 -5

0

-0.5 -5 log10 krot

0.8

0

-50

0

50

0 (deg)

(c) Mean FZ and FX vs. krot (

log10 krot o

0

=0 ) 0.6

Lift Thrust

0.6 0.4 0.2 0 -0.2 -5

-4

-3 -2 log10 krot

Nm

-1

-50

(d) Mean FZ and FX vs.

0

0

0 (deg)

50

(krot = kop)

0.2 0

-0.2 kop= 0.00392

0

0.4

0.5 (g-Force)

(g-Force)

Fig. 7 a Mean F Z vs. krot and ψ0 . b Mean F X vs. krot and ψ0 . c Mean values of F Z and F X vs. krot when ψ0 = 0◦ . At krot = kop = 3.92 × 10−3 N· m/rad, two wings are able to produce just enough lift to levitate the model. d Mean values of F Z and F X vs. ψ0 when krot = kop . All force values are normalized by the weight of the model. Stroke profile is always sinusoidal with constant parameters: φ0 = 60◦ and fs = 25 Hz

Figure 7a, b illustrate the changes in mean F Z and F X of a single wing as various krot and ψ0 pairs are examined. All force values are normalized by the weight of the model. At ψ0 = 0◦ , instantaneous forward thrust has an approximately zero mean value regardless of the selected krot (Fig. 7c). On the other hand, mean value of lift significantly varies as krot is changed. Particularly, when krot = kop = 3.92 × 10−3 N· m/rad, two wings will be able to produce just enough lift to levitate the model – i.e. 0.5 g-Force per wing. Therefore, we choose this point as the nominal mechanical impedance values for hovering. Note that when krot = kop , small changes in ψ0 slightly influence mean lift but also result in significant nonzero values of mean thrust (Fig. 7d). From Fig. 7c, d, around krot = kop and ψ0 = 0◦ , average lift and thrust are respectively proportional to stiffness and equilibrium angle ψ0 of the joint. Using these relationships, the Tunable Impedance method enables the model to regulate

Lift Thrust

/rad 0

-0.4 -60

-40

-20 0

0 20 (deg)

40

60

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aerodynamic forces during various agile flight maneuvers [23]. However, Fig. 7 also demonstrates that lift and thrust are influenced by both mechanical impedance parameters. Therefore, an attempt to control one force through adjustment of the corresponding mechanical impedance parameter may result in perturbation of the other force, thus complicating simultaneous control of both forces. In Section 4 we will investigate possible conditions that reduce the coupling between lift and thrust forces, hence allowing us to employ simpler control structures. The present work also concentrates on flight efficiency and power requirements of the Tunable Impedance approach. To do so, our proposed method will be compared to a conventional ad hoc control approach that manipulates frequency and shape of the stroke profile to adjust production of aerodynamic forces. From Eqs. 13–15, the magnitude of overall instantaneous aerodynamic force is approximately a quadratic function of fs . This relationship can also be observed in Fig. 8a where changes in mean lift and thrust vs. fs are plotted for a single wing. Note that in this case, mechanical impedance properties are always constant, i.e. krot = kop and ψ0 = 0◦ . From this diagram, stroke frequency can be used to control lift. However, mean thrust is unaffected by changes in fs . Note that as long as ψ0 = 0◦ and φ has a perfect sinusoidal waveform, thrust profile over one stroke cycle remains

4 Decoupling Conditions for Lift/Thrust Control To investigate the possibility of decoupled lift/thrust control via tunable impedance approach, we first revisit the diagrams in Fig. 7. From Fig. 7a and d, small changes in the value of ψ0 have little influence on overall lift. Similarly, Fig. 7b, c show that small changes in krot around kop only

(a) Average FZ and FX vs. fs

(b) Average FZ and FX vs. Tds

1.4 1.2

1.5 Lift Thrust 1

1 0.8

(g-Force)

(g-Force)

Fig. 8 a Mean values of F Z and F X vs. fs when Tds = 0.5. At fs = 25 Hz, two wings are able to produce just enough lift to levitate the model. b Mean values of F Z and F X vs. Tds when fs = 25 Hz. All force values are normalized by the weight of the model. Mechanical impedance parameters are always constant: krot = kop and ψ0 = 0◦

odd-symmetric [23]. Therefore, mean thrust will be close to zero. One way of producing nonzero thrust values is to employ asymmetric stroke profiles. This method is also known as Split Cycle [15]. Normally, upstroke and downstroke in one cycle take the same amount of time, hence their individual mean thrusts are equal and in opposite directions. By increasing the flapping speed in one part and reducing it in the other one, it is possible to disturb this balance and create forward or backward thrust. To this end, we define Tds as the ratio of time spent in downstroke to duration of the full stroke cycle. By definition, Tds = 0.5 is equivalent to a symmetric stroke profile and zero mean thrust. Figure 8b shows that a 20 % change in this value slightly increases mean lift, but at the same time, a mean thrust of approximately 0.25 g-Force per wing will be achievable. Hence, simultaneous manipulation of fs and Tds can be used as a direct approach to force/motion control in flapping-wing MAVs.

0.6 0.4 0.2

0.5

0

-0.5 Lift Thrust

0 -0.2 10

20

30 fs (Hz)

40

-1 0.2

0.4

0.6 Tds

0.8

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slightly affect overall thrust. Hence, as a trade-off between these characteristics and production of sufficiently large forces, we have chosen to limit the mechanical impedance parameters such that krot ∈ [0.2, 2] × 10−2 N·m/rad and |ψ0 | ≤ 20◦ . The force values in Fig. 7 are calculated for constant impedance parameters. However, the linear relationships between steady-state values suggest that a linearized model around krot = kop and ψ0 = 0◦ will be able to provide a reasonable approximation of dynamic behavior of these forces in response to varying impedance properties. To find this linear model, we employ system identification techniques—computation/smoothing of Empirical Transfer Function Estimates followed by subspace identification method [35] for transfer function matching—in order to estimate the components of nonlinear MIMO system that relates impedance parameters to aerodynamic forces: δ F Z (s) = G Z K (s) δ K (s) + Z (s)

(26)

δ F X (s) = (1 + X K (s)) G X (s) ψ0 (s)

(27)

δ F Z and δ F X respectively represent the changes in average lift and forward thrust due to variations of krot and ψ0 values around krot = kop and ψ0 = 0◦ . Note that δ K = log10 (krot / kop ). G Z K is the transfer function that relates δ K to average lift while G X represents the transfer function from ψ0 to average thrust. The impact of ψ0 varia-

(a)

(b) 0.2 Average FX (g-Force)

0.7 Average FZ (g-Force)

Fig. 9 a Average lift for impedance values around krot = kop and ψ0 = 0◦ . b Average thrust for impedance values around krot = kop and ψ0 = 0◦ . All force values are normalized by the estimated weight of the vehicle. Note that δ K = log10 (krot /kop )

tions on average lift and stiffness variations on average thrust are modeled by disturbance terms Z and X K , respectively. Figure 9 illustrates a closer look at Fig. 7a, b around the chosen operation point. These diagrams support our particular choice of disturbance terms in Eqs. 26 and 27. In all system identification experiments, limits of |δ K| < 0.05 and |ψ0 | < 0.1 rad have been enforced. The Bode diagrams of identified transfer functions are plotted in Fig. 10. Figure 11 illustrates the frequency profile of maximum gain from each mechanical impedance parameter to the corresponding disturbance term. In low frequencies, both G Z K and G X have steady-state gains close to 0 dB, while in comparison, the maximum gains in Fig. 11 are well below unity. Therefore, we expect that simultaneous manipulation of mechanical impedance parameters in the low frequency range has a less perturbing effect on individual lift and thrust control mechanisms. To choose a suitable upper limit for this range, three groups of numerical experiments have been conducted. In the first group, sinusoidal waves of magnitude 0.05 and various frequencies were used for δ K while ψ0 was set to 0 rad. In this case, changes in average lift and thrust are respectively treated as ideal F Z i and disturbance F Xd . In the second group, sinusoidal waves of magnitude 0.1 rad and various frequencies were used for ψ0 while krot was kept at kop . In this case, changes in average lift and thrust are respectively treated as disturbance F Z d and ideal F Xi . Finally, the

0.6 0.5 0.4 0.3 0.2 -0.2

10 K

0

0 0.2 -10

0

(deg)

0.1 0 -0.1 -0.2 0.2

-10 K

0

0 -0.2 10

0

(deg)

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(a) GZK -20

-20

Phase (deg)

Mag (dB)

0

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10

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150 100 50 -2 10

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(a) Δ ZΨ (s)/Ψ 0(s)

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Mag (dB)

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10

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(b) Δ XK(s)/δ K(s)

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0

0

10 10 Frequency (Hz)

From Fig. 12, when the frequency content of both inputs is below 10 Hz, the correlation between ideal and overall profile of either force is above 0.917 while the correlation between disturbance and overall values remains below 0.239. Thus, by employing low-pass filters to remove the frequency content of δ K and ψ0 above 10 Hz, coupling between lift and thrust manipulation could be reduced significantly.

-30

-100 -2 10

10

200

third group of experiments uses both described nonzero inputs at the same time. Lift and thrust changes in this group are the overall results of impedance manipulation, i.e. F Z o and F Xo . The frequency profiles of correlation between ideal (disturbance) and overall lift outputs are plotted in Fig. 12a, b. Similarly, frequency profiles of correlation between ideal (disturbance) and overall thrust outputs are plotted in Fig. 12c, d.

Mag (dB)

10

150

50 -2 10

Fig. 11 Frequency profile of maximum gain a from ψ0 to lift disturbance Z and b from δ K to thrust disturbance X K

(b) GXΨ

0

Phase (deg)

Mag (dB)

Fig. 10 Bode diagrams of the identified transfer functions a from δ K to average lift, i.e. G Z K and b from ψ0 to average thrust, i.e. G X

-100 -2 10

0

2

10 10 Frequency (Hz)

10

4

J Intell Robot Syst Fig. 12 Correlation coefficient vs. frequency of mechanical impedance manipulation for a F Z i and F Z o , i.e. ideal and overall lift, b F Z d and F Z o , i.e. disturbance and overall lift, c F Xi and F Xo , i.e. ideal and overall thrust, d F Xd and F Xo , i.e. disturbance and overall thrust. Here, f

and f K respectively represent the frequency of employed sinusoidal signals for ψ0 and δ K

(c) (FXi , FXo )

(a) (FZi , FZo ) 1

1

0.5

0.5

0 0

0

0

10

10 f (Hz)

10

20 f (Hz) K

20

10

f (Hz) 20

(b) (FZd, FZo )

20 f K (Hz)

(d) (FXd , FXo )

1

1

0.5

0.5

0

0 20 f

(Hz)

10

10 fK (Hz)

0 0

20

20

20

5 Dynamic Model of the Flapping-Wing MAV

f

(Hz)

(a) Frontal View

0 0

R

r

z

R

l

CoP

Fr

10 f K (Hz)

10

This body structure is inspired the proposed vehicle in [9]. The shape of each wing is as demonstrated in Fig. 5 while its other characteristics are as reported in Table 1.

The free-body diagram of a typical two-winged flapping-wing MAV model is illustrated in Fig. 13.

Fig. 13 Free-body diagram of a two-winged flapping-wing MAV: a frontal and b overhead views. H, R and U are the distance components of center of pressure (CoP) of each wing from center of mass (CoM) of the model. For simulation purposes, Euler angles in Tait-Bryan XYZ convention are employed to update the orientation of the body

0 0

CoP

Fl

Y

Y

Fr Z

H

r

H

l

Roll

Fl Z

Pitch

(b) Overhead View

y

Fr

x

Fl

X

X

CoM CoP U

Yaw

CoP

r

U

x

l

y

z

J Intell Robot Syst

Through employing Newton’s equations of motion, a six degree-of-freedom (DoF) mathematical model can be derived in this body frame [19]:    − b v |  × v = F l + F r − mbody G mbody v + ω v | v (28)  l × F l + D  r × F r − b ω ω ˙ + ω  × Jbody ω =D  Jbody ω (29)

⎡

where: ⎡

⎤ ⎡ r ⎤ l FX FX F l = ⎣ −FYl ⎦ , F r = ⎣ FYr ⎦ l r FZ FZ

and:

⎤ ⎡ r ⎤ Ul U  r = ⎣ −Rr ⎦  l = ⎣ Rl ⎦ , D D Hl Hr

Jbody (30)

⎤ 0 Jroll 0 = ⎣ 0 Jpitch 0 ⎦ 0 0 Jyaw

(32)

where Jroll , Jpitch and Jyaw are the body’s individual moments of inertia around x, y and z axes, respectively. All related physical parameters of the body are listed in Table 2.

⎡

(31)

Here, translational and angular velocities of the MAV’s body are represented by vectors v and ω, respectively. H, R and U are the distance components of center of pressure of each wing from the overall center of mass (CoM), as shown in Fig. 13. Superscripts r and l respectively refer to the right and left wings. The vector G expresses gravity in the body coordinate frame. To keep track of the orientation of the body with respect to the reference coordinates, Euler angles in the Tait-Bryan XYZ convention are employed. The

Table 2 Physical Characteristics of the modeled MAV’s body

coefficient of viscous friction is represented by b v while the parameter b ω is used to model the effect of passive rotational damping, as discussed in [36]. Mass of the body, i.e. mbody , is assumed to be 4 × 10−3 kg for a vehicle of desired scale (Table 1). The body shape in Fig. 13 is chosen since its corresponding inertia matrix relative to the center of mass (CoM), i.e. Jbody , has insignificant nondiagonal terms and thus, can be approximated by:

6 Flight Controllers Based on the discussions in Section 3.2, two different controllers are developed to control the simulated flight of dynamic model described in Section 5. As it was noted earlier, motion of the model will be restricted within sagittal plane, hence both controllers only deal with regulation of lift and forward thrust forces. In the remainder of this section, we will review the details and structure of these controllers.

Symbol

Description

Value

bω bv

Passive damping coefficient of the body (rotation) Viscous friction coefficient of the body when moving in the air Pitch and roll moments of inertia of the body relative to CoM Yaw moment of inertia of the body relative to CoM Distance of CoP from axial plane of the body (xy in Fig. 13) when ψ = 0◦ Distance of CoP from sagittal plane of the body (xz in Fig. 13) when φ = 0◦ Distance of CoP from coronal plane of the body (yz in Fig. 13) when φ = 0◦ Body width at the base of wings

3 × 10−3 N· m· s 4 × 10−4 N· m−2 · s2

Jpitch = Jroll Jyaw H(ψ = 0◦ ) R(φ = 0◦ ) U(φ = 0◦ ) Wbody

4.38 × 10−6 N· m· s2 1.15 × 10−7 N· m· s2 2.89 × 10−2 m 5.78 × 10−2 m 5.8 × 10−3 m 1.16 × 10−2 m

J Intell Robot Syst Fig. 14 Block diagram of the proposed Tunable Impedance motion controller in interaction with MAV model. Cutoff frequency f c of each low-pass filter is set to 10 Hz. Both wings employ similar values of β, krot and ψ0 at all times. The stroke profile properties are always constant: i.e. φ0 = 60◦ , fs = 25 Hz and Tds = 0.5

ref pitch

=0

o

-+

PD Pitch Controller

LPF with fc = 10 Hz

LQR Lift Controller

LPF with fc = 10 Hz

Z ref +

-

. Z

Z

= 60o fs = 25 Hz 0

K

Sinusoid Wave Generator

X ref

-+

LQR Thrust Controller

LPF with fc = 10 Hz

0

kop

. X

X

+

f(x)=10

x

×

krot

MAV Model pitch

6.1 Controller 1: Tunable Impedance Approach The block diagram in Fig. 14 shows how the Tunable Impedance controller is structured. To calculate appropriate LQR gains, the estimated transfer functions in Section 4 have been used to develop a linear model of the system from mechanical impedance parameters to vertical/horizontal position and velocity. The final result is as follows: δ K = 25 (Z − Z ref ) + 2 Z˙

(33)

˙ ψ0 = 2000 (X − Xref ) + 100 X

(34)

Fig. 15 Block diagram of controller 2 in interaction with MAV model. Both wings employ similar values of β, fs and Tds at all times. The magnitude of stroke and mechanical impedance parameters are always constant: i.e. φ0 = 60◦ , krot = kop and ψ0 = 0◦

ref pitch

where X and Z are respectively the horizontal and vertical positions of the model in sagittal plane. Xref and Z ref define the reference trajectory of motion within this plane. To reduce coupling (Fig. 7 c, d), the outputs of these controllers are chosen to be limited such that krot ∈ [0.2, 2] × 10−2 N·m/rad and |ψ0 | ≤ 20◦ . As argued in Section 4, presence of low-pass filters on the controller outputs further improves decoupling [37]. Note that for controller 1, the stroke profile properties of both wings are always constant: φ0 = 60◦ , fs = 25 Hz and Tds = 0.5. Pitch angle of the body θpitch tends to be unstable. To keep this angle small, i.e. to maintain the

= 0o PD Pitch Controller

+

-

LPF with fc = 10 Hz

Z ref +

-

X ref +

X

fs

Lift Controller (Gain)

. Z

Z

. X

0

Thrust Controller (Gain)

Function Generator

Tds

MAV Model pitch

= 60 o

+

J Intell Robot Syst

upright orientation of the model, we chose to bias the stroke profile with a variable amount represented by β. The details of this control mechanism are discussed in [23]. The value of β is adjusted by a PD sub-controller which is tuned to 10 + 0.05s through trial and error. The output of this controller is limited such that |β| ≤ 15◦ .

respectively controlled through changes in fs and Tds . The pitch sub-controller in this schematic is identical to the one described earlier for controller 1. The outputs of other sub-controllers are calculated as:

6.2 Controller 2: Split Cycle and Frequency Manipulation

(35)

˙ Tds = 0.5 − 2.5 (X − Xref ) − 0.2 X

(36)

and are limited to fs ∈ [14.3, 28.5] Hz and Tds ∈ [0.4, 0.6]. The gains in Eqs. 35 and 36 are tuned via trial and error such that in both control methods, simulated responses of the MAV to similar references remain close. The limits on fs and Tds are imposed to restrict achievable values of

Figure 15 demonstrates the block diagram of the second controller. In this approach, mechanical impedance parameters are always constant: krot = kop and ψ0 = 0◦ . Here, lift and forward thrust are

(a) Displacement along X-axis

(b) Displacement along Z-axis 1

XRef XActual

0.5

Z (m)

X (m)

1

ZActual

0 0

5

10

15

0

(c) Pitch Equilibrium of the Wings

6

5

10

15

-3 x 10 (d) Stiffness of the Joints

-10

N.m

krot (

0

0

(deg)

10

4 2

15

0

5

10

15

Time (s)

(e) Pitch Angle of the Body

2

pitch

5

(g) Trajectory in XZ plane 0 -2

1 0.8 0

5

10

15

(f) Stroke Bias Angle of the Wings

10 8

0.6

Reference Actual

0.4 0.2

6

0

4 2

Z (m)

0

(deg)

/rad)

10

ZRef

0.5

0

(deg)

Fig. 16 Simulated results for tracking a square trajectory using controller 1: a displacement along X and b Z axes, c pitch equilibrium of the wings ψ0 , d stiffness of the joints krot , e pitch angle of the body θpitch , f stroke bias angle of the wings β and g overall trajectory in XZ plane

fs = 25 − 187.5 (Z − Z ref ) − 12.5 Z˙

0

5

Time (s)

10

15

0

0.2

0.4 0.6 X (m)

0.8

1

J Intell Robot Syst

aerodynamic forces within the same range as controller 1 (Fig. 8a, b). The output of the function generator in Fig. 15 is originally a sinusoid of constant magnitude φ0 = 60◦ and variable frequency fs . Depending on the value of Tds , this waveform is further manipulated such that two consecutive half-cycles have appropriate durations.

7 Simulated Experiments The results of several simulated flight experiments will be discussed in this section. In the first group, we only employ controller 1 to investigate the performance of Tunable Impedance approach and

7.1 Decoupled Lift/Thrust Control via Tunable Impedance Approach Using controller 1, motion of the model along various trajectories in the XZ plane has been simulated. In each one of these experiments, the model is initially hovering at X = 0 m and Z = 0 m, i.e. Xref = 0 m and Z ref = 0 m. Later, new reference position-time profiles are introduced to sub-controllers that guarantee a smooth velocity

(a) Displacement along X-axis

(b) Displacement along Z-axis 1

XRef XActual

0.5

Z (m)

X (m)

1

ZRef ZActual

0.5

0

0 0

5

10

0

5

10

-3

(c) Pitch Equilibrium of the Wings 10

x 10 (d) Stiffness of the Joints

N.m

(deg)

/rad)

6

0

krot (

0

-10

0

5

4

2

10

0

5 Time (s)

(e) Pitch Angle of the Body

10

pitch

(deg)

2

(g) Trajectory in XZ plane 0

1

Reference Actual

0.8 0

5

10 Z (m)

-2

(f) Stroke Bias Angle of the Wings 10 (deg)

Fig. 17 Simulated results for tracking a linear trajectory with a slope of 1 using controller 1: a displacement along X and b Z axes, c pitch equilibrium of the wings ψ0 , d stiffness of the joints krot , e pitch angle of the body θpitch , f stroke bias angle of the wings β and g overall trajectory in XZ plane

its potential for decoupled control of lift/thrust forces. In the second group, experiments are repeated with both controllers. Overall performance and energy consumption in both cases can then be compared.

0.6 0.4

8 0.2

6

0

4 2

0 0

5 Time (s)

10

0.2

0.4 0.6 X (m)

0.8

1

J Intell Robot Syst

profile over desired trajectory. We observed that as long as acceleration demands of these profiles are within the capabilities of the model—primarily due to enforced limits on δ K and ψ0 —tracking will be achieved with a considerably high degree of precision. The results of simulated experiments with three important types of trajectories are presented next.

tracking a square trajectory that requires a combination of all these maneuvers, i.e. takeoff, landing and forward/backward motion. As it is illustrated, controller 1 successfully handles such maneuvers and keeps the model close to reference trajectory (Fig. 16g). Note that during each sub-maneuver, only one of the mechanical impedance parameters experiences significant changes while the other one remains almost constant (Fig. 16c, d).

7.1.1 Strictly Horizontal (Vertical) Maneuvers As the most trivial group of maneuvers, the MAV must be able to move along one axis while maintaining a constant position with respect to the other axis. Figure 16 demonstrates the results of

To further investigate the effectiveness of our proposed approach in decoupling lift and thrust

(a) Displacement along X-axis

(b) Displacement along Z-axis 1

XRef

Z (m)

X (m)

1

XActual

0.5

ZRef ZActual

0.5

0

0 0

5

10

0

5

10

-3

(c) Pitch Equilibrium of the Wings 10

x 10 (d) Stiffness of the Joints

N.m

(deg)

/rad)

6

0

krot (

0

-10

0

5

4

2

10

0

5 Time (s)

(e) Pitch Angle of the Body

10

pitch

(deg)

2

(g) Trajectory in XZ plane 0

1 0.8 0

5

10 Z (m)

-2

(f) Stroke Bias Angle of the Wings 10

0.6

Reference Actual

0.4

8 (deg)

Fig. 18 Simulated results for tracking a partially circular trajectory using controller 1: a displacement along X and b Z axes, c pitch equilibrium of the wings ψ0 , d stiffness of the joints krot , e pitch angle of the body θpitch , f stroke bias angle of the wings β and g overall trajectory in XZ plane

7.1.2 Simultaneous Motion along Two Axes: A Linear Trajectory

0.2

6

0

4 2

0 0

5 Time (s)

10

0.2

0.4

0.6

X (m)

0.8

1

J Intell Robot Syst

control, different trajectories with displacements along both X and Z axes were examined. A simple example of such trajectories is a straight line with a slope of 1. Simulation results for the corresponding experiment are illustrated in Fig. 17. A combination of forward flight and takeoff motion starting at t = 1 s enables the model to reach X = 1 m and Z = 1 m by t = 3 s. The simulated MAV maintains this position until t = 4 s when the direction of motion is reversed. Through simultaneous descent and backward flight, the model returns to its original position at t = 6 s. All the while, the slope of its trajectory remains close to 1 (Fig. 17g).

Curves are a more complex case of trajectories with simultaneous displacement along both X and Z axes. The results of an experiment with a partially circular reference trajectory are plotted in Fig. 18. The model still manages to follow this trajectory with reasonable precision (Fig. 18g). Note that compared to other trajectories, correction of motion over curves requires slower changes in impedance parameters (Fig. 18c, d). This is partially due to the fact that stable motion along such trajectories is rarely accompanied by high velocities and/or acceleration rates.

(a) Displacement along X-axis

(b) Displacement along Z-axis

0.01

0.01 XActual

ZRef

Z (m)

X (m)

XRef

0

-0.01

0

0.5

1

-0.01

N.m

2 0

x 10 (d) Stiffness of the Joints

0

0.5

2

1

0

0

2

0

0.5

0.5

1

(g) Overall Energy Consumption 2.5

1

(f) Stroke Bias Angle of the Wings 10

ETI (J)

(deg)

(e) Pitch Angle of the Body

pitch

1

4

2

-2

0.5

/rad)

6

krot (

(deg) 0

0 -3

4

-2

ZActual

0

(c) Pitch Equilibrium of the Wings

1.5

1

8 (deg)

Fig. 19 Simulated results for hovering with controller 1: a displacement along X and b Z axes, c pitch equilibrium of the wings ψ0 , d stiffness of the joints krot , e pitch angle of the body θpitch , f stroke bias angle of the wings β and g overall energy consumption ETI

7.1.3 Movement Along Curves

6

0.5

4 2

0

0.5 Time (s)

1

0

0

0.5 Time (s)

1

J Intell Robot Syst

stroke axis while Jφ represents the wing’s moment of inertia along this axis. The calculated value for this parameter is equal to 4.894 × 10−7 N·m·s2 (Fig. 5 with wing mass = 2 × 10−4 kg). The energy required for both wings to follow a specific stroke profile, i.e. Estroke is then estimated by:  t τstroke φ˙ dtˆ Estroke = 2 (38)

7.2 Method Comparison: Performance and Energy Consumption Several simulated trajectory tracking experiments are chosen to be repeated with both controllers of Section 6. Some of these results are presented here to compare flight performance and power requirements. To estimate stroke power in both sets of experiments, stroke torque of each wing is calculated first: τstroke = F DrCoP + b φ φ˙ + Jφ˙ φ˙

0

Since actuation in the second control approach is only limited to changes in stroke profile, Eq. 38 is in fact an estimate of this method’s overall energy consumption. As for controller 1, changes in mechanical impedance properties require extra

(37)

Here, b φ = 1 × 10−5 N·m·s is the passive damping coefficient of each wing when rotating around its

(a) Displacement along X-axis

(b) Displacement along Z-axis

0.01

0.01 XActual

ZRef

Z (m)

X (m)

XRef

0

-0.01

0

0.5

1

ZActual

0

-0.01

0

(c) Downstroke Duration Ratio

0.5

1

(d) Stroke Frequency

0.6

27 fs (Hz)

Tds

26 0.5

0.4

0

0.5

25 24 23

1

0

0.5

1

(g) Overall Energy Consumption

2

2.5

0

2

-2

0

0.5

1

(f) Stroke Bias Angle of the Wings 10

Estroke (J)

pitch

(deg)

(e) Pitch Angle of the Body

1.5

1

8 (deg)

Fig. 20 Simulated results for hovering with controller 2: a displacement along X and b Z axes, c downstroke duration ratio Tds , d stroke frequency fs , e pitch angle of the body θpitch , f stroke bias angle of the wings β and g overall energy consumption Estroke

6

0.5

4 2

0

0.5 Time (s)

1

0

0

0.5 Time (s)

1

J Intell Robot Syst

power. Using Eq. 12, total energy consumption in this approach, i.e. ETI is estimated by:  t |PTI |dtˆ (39) ETI = Estroke + 2

the MAV model over a period of 1 s has been simulated. The results for controllers 1 and 2 are plotted in Figs. 19 and 20, respectively. From these diagrams, it is observed that both controllers are able to achieve stable hovering without requiring significant changes in their output values. Since all control outputs remain close to their nominal values, it can be concluded that: 1) body pitch profiles in both approaches are close and 2) the value of PTI for controller 1 is insignificant. As a result, both cases should have similar power requirements. In fact over a hovering period of 1 s, energy consumption with controllers 1 and 2 is estimated to be 2.2116 J and 2.2078 J, respectively (Figs. 19g and 20g). Energytime profiles in both cases are approximately

0

Based on dimensions of a medium-sized hummingbird, parameters A and R in Eq. 12 are set to appropriate values such that A2 R6 = 2.5 × 10−5 N2 ·m2 . 7.2.1 Hovering Perhaps, hovering is the only flight capability that is unique to insects and hummingbirds. Flappingwing MAVs with a similar stroke pattern are also capable of hovering. In this part, hovering of

(a) Displacement along X-axis

(b) Displacement along Z-axis

XRef

0.5

10 Z (m)

X (m)

XActual

0 -0.5

5 ZRef

5

10

15

0

(c) Pitch Equilibrium of the Wings

15

/rad) N.m

0.01

krot (

0

0

(deg)

10

0.02

-20 0

5

10

0

15

0

(e) Pitch Angle of the Body

5

10

15

(g) Overall Energy Consumption 35

2 (deg)

5

(d) Stiffness of the Joints

20

pitch

ZActual

0 0

30 0 25 0

5

10

15

(f) Stroke Bias Angle of the Wings

ETI (J)

-2 20 15

15 (deg)

Fig. 21 Simulated results for vertical takeoff and zigzag descent with controller 1: a displacement along X and b Z axes, c pitch equilibrium of the wings ψ0 , d stiffness of the joints krot , e pitch angle of the body θpitch , f stroke bias angle of the wings β and g overall energy consumption ETI

10

10

5 5

0 -5 0

5

10 Time (s)

15

0

0

5

10 Time (s)

15

J Intell Robot Syst

linear, suggesting a constant energy consumption rate of 2.2116 Watt and 2.2078 Watt, respectively. Note that these values are required to maintain levitation and do not represent cost of transportation. It is worth mentioning that hummingbirds spend up to 85 % of their time sitting and digesting [38]. The rest is spent for various flight activities that often involve foraging. In [39], it has been observed that a 4.3-gram Broad-tailed hummingbird consumes about 7.31 Calories of sugar per day. If the energy consumed at rest is ignored, this amount would be equivalent to an approximate in-flight energy consumption rate of 2.36 Watt which is close to the power requirements of our MAV model during hovering.

The results of a takeoff/descent maneuver with controllers 1 and 2 are respectively plotted in Figs. 21 and 22. In these experiments, the hovering model is initially commanded to vertically takeoff such that it reaches an altitude of 10 m within 4 s. After 1 s of hovering at this altitude, the model must descend to a height of 5 m within another 4 s. However, reference displacements along X and Z -axes are designed such that this part of the trajectory follows a zigzag pattern (Figs. 21a, b and 22a, b). Both employed control strategies are able to precisely navigate the model along this trajectory

(a) Displacement along X-axis

(b) Displacement along Z-axis

XRef

10

X (m)

XActual

0

Z (m)

0.5

-0.5

5 ZRef

5

10

15

0

(c) Downstroke Duration Ratio

fs (Hz)

Tds

10

15

30

0.5

25 20 15

0.4 0

5

10

15

0

(e) Pitch Angle of the Body

5

10

15

(g) Overall Energy Consumption 35

2 (deg)

5

(d) Stroke Frequency

0.6

pitch

ZActual

0 0

30 0 25 0

5

10

15

(f) Stroke Bias Angle of the Wings 15 10

Estroke (J)

-2

(deg)

Fig. 22 Simulated results for vertical takeoff and zigzag descent with controller 2: a displacement along X and b Z axes, c downstroke duration ratio Tds , d stroke frequency fs , e pitch angle of the body θpitch , f stroke bias angle of the wings β and g overall energy consumption Estroke

7.2.2 Vertical Takeof f Followed by Zigzag Descent

20 15 10

5 5

0 -5 0

5

10 Time (s)

15

0

0

5

10 Time (s)

15

J Intell Robot Syst

(Fig. 21a, b and 22a, b). Note that during takeoff with controller 1, the outputs of thrust and pitch sub-controllers are almost constant (t = 1 s to t = 5 s in Fig. 21c and f). The same is not true for controller 2 (Fig. 22c and f). Furthermore, compared to controller 1, it seems the thrust and pitch sub-controllers in controller 2 have to operate at higher frequencies in order to provide the same overall performance. Compared to hovering experiments, the model with controller 1 does not demonstrate considerable changes in terms of energy consumption. From Fig. 21g, average power requirements during takeoff and descent maneuvers are estimated as 2.1651 Watt and 2.1596 Watt, respectively. Note that these values are slightly less than power requirements during hovering. From Eq. 12, only a

(a) Displacement along X-axis

(b) Displacement along Z-axis

150

0.01 XActual

0

100

Z (m)

X (m)

XRef

50 0

0

5

10

15

-0.03

/rad) N.m

krot (

-10

0

(deg)

0

-20 5

10

0

15

6 4 2 0

15

0

5

10

15

(g) Overall Energy Consumption 35

40 (deg)

10

x 10 (d) Stiffness of the Joints

(e) Pitch Angle of the Body

pitch

5 -3

8

0

ZActual

-0.02

10

-30

ZRef

-0.01

(c) Pitch Equilibrium of the Wings

30 20 25 0

5

10

15

(f) Stroke Bias Angle of the Wings 20

ETI (J)

0

(deg)

Fig. 23 Simulated results for forward flight at 10 m· s−1 with controller 1: a displacement along X and b Z axes, c pitch equilibrium of the wings ψ0 , d stiffness of the joints krot , e pitch angle of the body θpitch , f stroke bias angle of the wings β and g overall energy consumption ETI

small fraction of overall power (less than 1 %) is spent for impedance manipulation (Fig. 21c, d). The rest, i.e. Estroke , is used to maintain a sinusoidal stroke profile with constant magnitude and frequency. Therefore, from Eqs. 37 and 38, β and F D are the only variables that can affect average stroke power requirements. Changes in β are often small and slow (Fig. 21f). As for F D , it has been previously shown [23] that around kop , smaller values of stiffness decrease the magnitude of this force. Hence, the slight reduction in overall value of Estroke (and ETI ) in this experiment is primarily due to changes in stiffness (Fig. 21d). Unlike controller 1, controller 2 relies on modification of stroke properties. Thus, Eqs. 37 and 38 imply that power requirements in experiments with this controller will depend on desired

20 15

15

10

10

5

5

0

5

10 Time (s)

15

0

0

5

10 Time (s)

15

J Intell Robot Syst

maneuver. From Fig. 22g, these requirements during takeoff and descent maneuvers are estimated as 2.3164 Watt and 2.1876 Watt, respectively. The power used for takeoff is approximately 5 % more than hovering requirements. Since necessary lift for takeoff is achieved through faster stroke cycles (Fig. 22d), the increase in energy consumption is to be expected. As for descent, a similar argument can be used to justify the slight reduction in power requirements compared to those of hovering mode.

thrust boost necessary for such velocities is provided through forward pitching of the body. The downside to this strategy is that it also reduces the maximum possible value of lift; hence, vertical maneuvers at such velocities are very challenging. The presented flapping-wing MAV model also demonstrates such behavior with both controller 1 and 2. Here, the model can reach velocities as high as 10 m·s−1 before lift sub-controllers fail to stabilize its altitude. The simulated results with controller 1 and 2 at 10 m·s−1 are illustrated in Figs. 23 and 24, respectively. In both cases, the strained performance of lift sub-controller results in a slight altitude loss (Figs. 23b and 24b). From Fig. 23g, the model with controller 1 requires only 2.0018 Watt to maintain its target velocity. This value is 9.5 % lower than the power

7.2.3 Forward Flight at High Velocities Although average flight speed of hummingbirds is about 5–7 m·s−1 , they are known to be able to reach velocities as high as 14 m·s−1 [40]. The

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required for hovering. Similar to the takeoff experiment in Section 7.2.2, this reduction is caused by lowered values of stiffness (Fig. 23d). The model with controller 2 reaches target velocity through employing higher values of stroke frequency (Fig. 24d). From Fig. 24g, this approach requires 2.8541 Watt which is 29.3 % larger than the power required for hovering. Note that for the same maneuver with controller 1, required power is reduced by 0.8523 Watt, i.e. almost 30 %.

8 Conclusions Inspired by insect/hummingbird flight, tunable impedance has been proposed as a semi-passive approach to force manipulation in flapping-wing MAVs. This method states that pitch rotation profile of each wing can be modified through changes in mechanical impedance properties of its joint (Fig. 6). Consequently, as shown by Eq. 25, magnitude of variations and average value of this profile are influential factors in adjustment of aerodynamic forces. Therefore, regulation of mechanical impedance properties can be used as an indirect means of controlling these forces (Fig. 7). The primary advantage of TI over conventional methods of force control is that it requires no modification in stroke profile of either wing. In fact, both wings can use the same sinusoidal pattern of flapping with fixed parameters – except for stroke bias angle β which is an independent control parameter. This effectively simplifies the control system and can reduce the amount of onboard hardware in an actual prototype. Under this approach, the system is also more controllable and has a higher degree of mobility [26]. However, lift and thrust control are not completely decoupled which at times may result in limited maneuverability of the vehicle. In this work, we showed that in a system with TI as its underlying strategy for force control, coupling between lift and thrust effectively occurs when either impedance variable carries significant components at high frequencies, i.e. close to the wing-beat rate (Fig. 12). Removing high frequency content from both impedance parameters will considerably reduce lift/thrust coupling and results in better performance compared to

our previous work [26]. In fact, a simple control structure such as Fig. 14 is sufficient to stabilize motion along various trajectories. Results of simulated experiments in Section 7 verify the tracking capabilities of this method. It has been observed that as long as a trajectory can be described using smooth velocity profiles, the only concern for successful tracking is the corresponding acceleration requirements. While imposed limits on impedance parameters guarantee linear relationships with forces, they also restrict acceleration capabilities of the model (Fig. 7). Thus, successful tracking also requires that timederivatives of these velocity profiles lie within acceleration limits of the model. In a real system, designing such profiles relies on information from both environment and vehicle itself. A high level controller must first come up with an appropriate trajectory based on surroundings of the vehicle and relative location of its target. The current velocity and acceleration limits of the MAV must then be treated as constraints of designing a suitable velocity profile for tracking purposes. The second goal of this work was to investigate the impact of Tunable Impedance approach on power requirements of a flapping-wing MAV. It was observed that the necessary amount of energy for impedance manipulation is a lot smaller than the amount required for stroke motion. Since stroke profile is approximately constant, the total power requirement in this approach is almost independent of desired maneuver. For the simulated MAV model in this paper, the required power is always less than 2.21 Watt. Interestingly, this value is in agreement with average in-flight energy consumption rate of a hummingbird with similar body dimensions. In comparison to Tunable Impedance, other control approaches that rely on manipulation of stroke properties often require considerably greater amounts of power. This increase in power consumption is specifically observed when a maneuver demands more lift or thrust production (e.g. forward acceleration in Fig. 24). Faster stroke motions are usually employed to provide for this excess in force requirements. The gap in energy consumption will widen further when we consider the case in which pitch rotation of the wing is not governed by a passive

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mechanism. Consequently, additional actuators are required to directly rotate the wing and modify its angle of attack throughout each stroke cycle. From Eqs. 14 and 24, one can estimate the active torque τψ = τext that must be applied to each wing so that it follows a predefined pitch profile ψ. The energy required for active pitch rotation of both wings, i.e. Eψ is then calculated as:  t τψ ψ˙ dtˆ Eψ = 2 (40) 0

From Eq. 40, if the model with controller 2 were to actively generate the same wing pitch profiles as the simulations in Section 7.2, its power requirements would increase by 0.5571 Watt during hovering, 0.6613 Watt during takeoff and 0.9540 Watt when moving forward at 10 m·s−1 . As a bio-inspired control strategy with low power demands, the Tunable Impedance method is a promising solution to both problems of maneuverability and power efficiency in flappingwing MAVs. Considering the limitations of available batteries, implementation of this approach may significantly improve flight time. Our future work is focused on development of appropriate actuators for impedance manipulation. Above all, such actuators must have minimized loading effects so that wing’s dynamics are not greatly perturbed. Upon achieving satisfactory results in impedance manipulation, our eventual goal is to install the designed actuators on a 3-inch-wingspan flapping-wing MAV and conduct actual flight experiments.

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